Twisting theory, relative Rota-Baxter type operators and $L_\infty$-algebras on Lie conformal algebras

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct $L_\infty$-algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a $\mathbb{C}[\partial]$-module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the $\mathbb{C}[\partial]$-module homomorphism is a Maurer-Cartan element of the $L_\infty$-algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations.